(i) Since \(\mathcal {J}_n\)  is isomorphic to \(\mathbb {R}^{k_n}\) , and thus \(\mathcal {J}_n^*\)  is isomorphic to \(\mathbb {R}^{k_n}\) . By [1]} Lemma 2.6.15 and Lemma 2.6.18(vii), the VC index of \(\mathcal {G}_n\)  satisfies \(V(\mathcal {G}_n) \le (k_n + 2)^2\) .
